Taylor M.E. Partial Differential Equations III: Nonlinear Equations

678 18. Einstein’s Equations

Hence

(8.7) x

`

D 

`

:

In other words, if the coordinates x

j

satisfy x

j

D 0 (we call them “harmonic

coordinates”), then 

`

D 0. Thus, in a harmonic coordinate system, Ric

jk

is equal

to

(8.8)

e

R

jk

D

1

2

g

`m

@

`

@

m

g

jk

C H

jk

.g; rg/:

At this point we can solve the initial-value problem

(8.9)

g

`m

@

`

@

m

g

jk

 2H

jk

.g; rg/ D 0;

g

jk

ˇ

S

D

ı

g

jk

;@

0

g

jk

ˇ

S

D

ı

k

jk

;

as long as

ı

g

jk

deﬁnes a Lorentz inner product on T

p

M ,forwhichT

p

S is space-

like, for each p 2 S. In such a case, this is a quasi-linear hyperbolic system, to

which the results of Chap. 16,  3, apply. We will also have a solution to (8.1), if

we can show that 

j

D 0. To that end, we establish the following:

Lemma 8.1. If

e

R

jk

D 0,then

j

satisﬁes a system of PDE of the form

(8.10) @

k

@

k



j

C B

j`

m

.g; rg/ @

`



m

D 0:

Proof. By (8.4)and(8.8), if

e

R

jk

D 0,then

Ric

jk

D

1

2

g

j`

@

k



`

C

1

2

g

k`

@

j



`

;

and hence

(8.11) G

jk

D

1

2



@

k



j

C @

j



k

 g

jk

@

`



`



:

Thus

2G

jk

Ik

D



@

k



j

C @

j



k



Ik

 g

jk

.@

`



`

/

Ik

;

and a straightforward calculation gives

(8.12) 2G

jk

Ik

D @

k

@

k



j

C B

j`

m

.g; rg/ @

`



m

:

Since G

jk

Ik

D 0,wehave(8.10).

8. The initial-value problem 679

We note for future reference that, without the assumption that

e

R

jk

D 0,we

have

(8.13) @

k

@

k



j

C B

j`

m

.g; rg/ @

`



m

D2

e

T

jk

Ik

;

e

T

jk

D

e

R

jk



1

2

e

R

`

g

jk

:

Now, (8.10) is a linear hyperbolic system for 

j

. We can deduce that 

j

D 0

on a neighborhood of S if we have

(8.14) 

j

ˇ

S

D 0; @

0



j

ˇ

S

D 0:

Arranging this requires placing appropriate compatibility conditions on the

Cauchy data (8.2). We now turn to this.

From (8.5)wehave

(8.15) 

`

ˇ

S

D

ı

k

`0



1

2

ı

g

`0

ı

k

j

C F

`

.

ı

g/r

S

ı

g;

where the last term is linear in r

S

ı

g D .@

1

ı

g; @

2

ı

g; @

3

ı

g/. Also, from (8.11)wesee

that

(8.16) 2G

0

k

D @

k



0

C g

k`

@

0



`

 ı

0

k

@

`



`

;

provided

e

R

jk

D 0. Consequently,

(8.17)

e

R

jk

D 0; 

`

ˇ

S

D 0 H) 2G

0

k

ˇ

S

D

ı

g

k`

@

0



`

ˇ

S

:

At this point it is convenient to record the following observation:

Lemma 8.2. The restriction G

0

k

ˇ

S

is given in terms of

ı

g

jk

;

ı

k

jk

, and their tan-

gential derivatives; it does not involve @

2

0

g

jk

.

Proof. From (8.3)wehave

(8.18)

G

0

k

D

1

2

g

0j

g

`m

.@

k

@

m

g

`j

C @

`

@

j

g

km

 @

`

@

m

g

jk

 @

j

@

k

g

`m

/



1

2

ı

0

k

g

ji

g

`m

.@

i

@

m

g

`j

 @

`

@

m

g

ji

/ C

e

H

k

.g; rg/:

The contribution of the terms involving @

2

0

is 1=2 times

g

0j

g

`0

ı

0

k

@

2

0

g

`j

C g

00

g

0m

@

2

0

g

km

 g

0j

g

00

@

2

0

g

jk

 ı

0

k

g

00

g

`m

@

2

0

g

`m

.ı

0

k

g

j0

g

`0

@

2

0

g

`j

 ı

0

k

g

ji

g

00

@

2

0

g

ji

/;

which clearly vanishes.

680 18. Einstein’s Equations

Let the resulting formula for G

0

k

ˇ

S

be

(8.19) G

0

k

ˇ

S

D G

0

k



ı

g

jk

;D

2

S

ı

g

jk

;

ı

k

jk

; r

S

ı

k

jk



:

We now state a local existence result:

Proposition 8.3. Suppose the initial data (8.2)areC

1

on S and satisfy the con-

sistency condition

(8.20) G

0

k



ı

g

jk

;D

2

S

ı

g

jk

;

ı

k

jk

; r

S

ı

k

jk



D 0:

Assume S is spacelike for

ı

g

jk

. Then the initial-value problem (8.1)–(8.2) has a

C

1

-solution on a neighborhood of S.

Proof. Start with the tensor ﬁeld

(8.21) eg

jk

.x/ D

ı

g

jk

.x

0

/ C x

0

ı

k

jk

.x

0

/; x

0

D .x

1

;x

2

;x

3

/;

which deﬁnes a Lorentz metric on a neighborhood of S. Then the Einstein tensor

e

G

jk

of this metric satisﬁes (8.19) and hence

e

G

0

k

ˇ

S

D 0.

Now deﬁne smooth “harmonic” coordinates y

0

;:::;y

3

, by solving

(8.22)

e

y

j

D 0;

with Cauchy data

(8.23) y

j

ˇ

S

D x

j

ˇ

S

;dy

j

ˇ

S

D dx

j

ˇ

S

;

where

e

u is the analogue of (8.6)forthemetriceg

jk

. Rewrite the initial-value

problem (8.1)–(8.2), in this new coordinate system.

Then (8.2) takes the form

(8.24) g

jk

.0; y

0

/ D

ı

g

jk

.y

0

/;

@

@y

0

g

jk

.0; y

0

/ D

Q

k

jk

.y

0

/I

the functions

ı

g

jk

are not changed, but the

ı

k

jk

do undergo a change. Due to the

tensor character of

e

G

jk

,wehave

(8.25) G

0

k

.

ı

g

jk

;D

2

S

ı

g

jk

;

Q

k

jk

; r

S

Q

k

jk

/ D 0:

Now solve the system (8.9)forg

jk

,inthe.y

0

;:::;y

3

/-coordinates, using the

initial data (8.24). We claim that .y

0

;:::;y

3

/ are harmonic coordinates for the

Lorentz metric g

jk

. In fact, by Lemma 8.1 it sufﬁces to show that if 

`

are given

8. The initial-value problem 681

by (8.5), then 

`

ˇ

S

D 0 and .@=@y

0

/

`

ˇ

S

D 0. Note that (8.22), together with

(8.15), implies that 

`

ˇ

S

D 0 when 

`

is determined by any metric g

jk

satisfying

(8.26) g

jk

eg

jk

D O.y

2

0

/:

Thus we have 

`

ˇ

S

D 0 in our case. Next, by (8.25) and Lemma 8.2, G

0

k

ˇ

S

D 0,

so (8.17) implies .@=@y

0

/

`

ˇ

S

D 0.

Since .y

0

;:::;y

3

/ are harmonic coordinates for the metric g

jk

,wehave(8.1)

as a consequence of (8.9). Converting back to .x

0

;:::;x

3

/-coordinates, we also

have (8.2), as a consequence of (8.26). This completes the proof.

For simplicity, we have not speciﬁed which Sobolev spaces are needed for

the initial data. Due to the special structure of Einstein’s equations, one can

obtain solutions with less regularity than is needed for general second-order,

quasi-linear hyperbolic systems. Results on this can be found in [HKM]andin

Chap. 5 of [Tay].

We have the following local uniqueness result:

Proposition 8.4. Suppose g

jk

and g

0

jk

are two smooth solutions to (8.1)–(8.2),

on a neighborhood of S . Then there exists a C

1

-diffeomorphism ' on a neigh-

borhood of S, such that '

ˇ

S

D id. and '



g D g

0

.

Proof. Without loss of generality, one can assume that the coordinates

.x

0

;:::;x

3

/ are harmonic for the metric g

jk

. Parallel to (8.22)–(8.23), solve



0

y

j

D 0; y

j

ˇ

S

D x

j

;dy

j

ˇ

S

D dx

j

,where

0

is the Laplace–Beltrami opera-

tor for the metric g

0

. Then the diffeomorphism '.y/ D x does the trick, since the

system (8.9)forg (in the x-coordinates) is precisely the same as the system for

g

0

(in the y-coordinates), and solutions to this quasi-linear hyperbolic system are

locally unique.

We have seen that one way to “hyperbolicize” the equation (8.1)istouse

harmonic coordinates. We now discuss an alternative method, due to D. DeTurck

[DeT]. In this method, (8.1) is modiﬁed to

(8.27) Ric.g/  div





W

1

div G.W /



D 0;

where W is a convenient second-ordersymmetric tensor ﬁeld, which we will spec-

ify below, and G acts linearly on S

2

T



,bytherule

(8.28) G.W /

jk

D W

jk



1

2

.Tr W/g

jk

:

In fact, if the initial data for g

jk

are given by (8.2), we set

(8.29) W

jk

.x/ D eg

jk

.x/ D

ı

g

jk

.x

0

/ C x

0

ı

k

jk

.x

0

/;

682 18. Einstein’s Equations

as in (8.21). Note that, upon lowering an index, W deﬁnes an invertible endomor-

phism on the tangent bundle T ; we denote the inverse by W

1

.

For any given invertible W , B D div



.W

1

div G.W // depends on the metric

tensor g

jk

; a calculation shows that it is given by

(8.30) B

jk

D

1

2

g

`m



@

j

@

k

g

`m

C @

m

@

k

g

`j

C @

`

@

j

g

mk



C C

jk

.g; rg/:

Comparison with (8.3)gives

(8.31) Ric

jk

 B

jk

D

1

2

g

`m

@

`

@

m

g

jk

C M

jk

.g; rg/  C

jk

.g; rg/:

Thus the equation (8.27) is strictly hyperbolic. Again, the results of Chap.16,  3,

apply. We now want to show that if the initial data (8.2) satisfy the compatibility

conditions of Proposition 8.3,thenB

jk

D 0, so again we get a solution to (8.1).

More precisely, we establish the vanishing of

(8.32) u D W

1

div G.W /:

In fact, applying div ı G to both sides of (8.27) and using G

jk

Ik

D 0,wehave

(8.33) div G div



u D 0;

when g satisﬁes (8.27). Note that, for any covariant vector ﬁeld v,

(8.34)

.div G div



v/

j

D

1

2

g

`m

.v

`ImIj

 v

j I`Im

 v

`Ij Im

/

D

1

2

.g

`m

v

j I`Im

 Ric

`

j

v

`

/;

so (8.33) is a strictly hyperbolic equation for u. Now, the construction (8.29)of

W guarantees that div G.W / D 0 on S Dfx

0

D 0g,so

(8.35) u

ˇ

S

D 0:

Thus, the vanishing of u would follow from @

0

u

ˇ

S

D 0. To get this, we use a

lemma:

Lemma 8.5. If v is a covariant vector ﬁeld on a Lorentz manifold .M; g/,then

(8.36) v

ˇ

S

D 0; G.div



v/

0

j

ˇ

S

D 0 H) @

0

v

ˇ

S

D 0:

Proof. We have

G.div



v/

0

j

D

1

2

.v

j

I0

C v

0

Ij

 ı

0

j

g

`m

v

`Im

/:

8. The initial-value problem 683

Hence

v

ˇ

S

D 0 H) G.div



v/

0

j

D

1

2

g

00

@

0

v

j

on S; for j D 1; 2; 3:

In particular, the hypotheses in (8.33) yield that @

0

v

j

ˇ

S

D 0,forj D 1; 2; 3.

Granted that, we see that, on S; g

`m

v

`Im

D g

00

v

0I0

,so

G.div



v/

0

ˇ

S

D

1

2

g

00

@

0

v

0

on S;

and this yields the complete implication (8.36).

Now, the compatibility conditions (8.20) on the initial data imply that

Ric

0

j

ˇ

S

D 0,soif(8.27) holds, then u D W

1

div G.W / satisﬁes all the

hypotheses of (8.36). Thus @

0

u

ˇ

S

D 0, so we have the following result:

Proposition 8.6. If the initial data satisfy the compatibility conditions (8.20),

then the solution to the hyperbolic system (8.27) is also a solution to (8.1).

Having examined the empty-space case, we next consider Einstein’s equations

coupled with Maxwell’s equations for an electromagnetic ﬁeld, (5.1). In parallel

with the approach to the empty-space case in (8.9), we will consider the system

(8.37)



1

2

ng

`m

@

`

@

m

g

jk

C H

jk

.g; rg/ D 8



T

jk



1

2

g

jk



;

F D 0;

where, in the ﬁrst equation,  D T

j

and

(8.38) T

jk

D

1

4



F

j

`

F

k`



1

4

F

i`

F

i`

g

jk



is the stress-energy tensor for the electromagnetic ﬁeld, as in (5.2). We have

obtained the second equation in (8.37) from d F D 0 D d

F

F,byusing

 Ddd

F

 d

F

d . In local coordinates, this operator depends on g

jk

and

derivatives up to second order; in fact,

(8.39) .F /

jk

D g

`m

@

`

@

m

F

jk

C E

jk

.D

2

g; rF /:

Let us pose initial data on a compact hypersurface S , including the data (8.2),

with

(8.40)

ı

g

jk

2 H

sC1

.S/;

ı

k

jk

2 H

s

.S/:

684 18. Einstein’s Equations

We assume that S is spacelike for these data and that s>7=2. We also specify

(8.41) F

jk

ˇ

S

2 H

s

.S/; @

0

F

jk

ˇ

S

2 H

s1

.S/:

We will postpone placing compatibility conditions on these data. If we identify a

neighborhood of S with I  S; I D .a; a/, then, for a sufﬁciently small, we

will obtain a solution to (8.37)–(8.41) satisfying

(8.42)

g 2 C



I;H

sC1

.S/



\ C

1



I;H

s

.S/



;

F 2 C



I;H

s

.S/



\ C

1

.I; H

s1

.S/



:

The local existence of solutions to (8.37) is established by a slight variant of the

method developed in  1–3 of Chap. 16 to treat hyperbolic systems. One obtains

ﬁrst-order systems for .'; /, with ' D .ƒg

jk

;@

0

g

jk

/ and D .ƒF

jk

;@

0

F

jk

/.

From there one solves approximating systems for .'

"

;

"

/ and uses energy esti-

mates plus Gronwall’s inequality to establish

(8.43) k'

"

.t/k

2

H

s

Ck

"

.t/k

2

H

s1

 C;

for jtj <a. We require that s>7=2,soH

s

.S/  C

2Cr

.S/ for some r>0.From

(8.43) it follows that a limit point exists, yielding a solution to (8.37)–(8.41). As

in Chap. 16, one establishes uniqueness, and the continuity described in (8.42).

The reason one uses different Sobolev estimates for '.t/ and for .t/ is the oc-

curence of second-order derivatives of g

jk

in (8.39), compensated by the fact that

no derivatives of F

jk

are involved in the ﬁrst equation of (8.37).

Now, assume that F

ˇ

S

and @

0

F

ˇ

S

satisfy the compatibility conditions

(8.44) d F D 0 D d

F

F on S:

Since d D d  and d

F

D d

F

,wehave

(8.45) .d F / D 0; .d

F

F/ D 0:

We deduce that d F D 0 D d

F

F on I  S. As discussed in  1, this implies that

T

jk

,givenby(8.38), satisﬁes

(8.46) T

jk

Ik

D 0:

We next want to show that if g

jk

ˇ

S

and @

0

g

jk

ˇ

S

satisfy appropriate com-

patibility conditions, then 

`

Dx

`

D 0,soinfacttheEinsteinequations

G

jk

D 8T

jk

follow from (8.37).

Lemma 8.7. Assume that we have a solution to (8.37)–(8.41)onI  S and that

(8.46) holds. Assume that 

`

ˇ

S

D 0,for0  `  3, and that, for 0  k  3,

8. The initial-value problem 685

(8.47) G

0

k

.

ı

g

jk

;D

2

S

ı

g

jk

;

ı

k

jk

; r

S

ı

k

jk

/ D 8T

0

k

:

Then 

`

D 0 on I  S .

Proof. If (8.39) holds, then (8.13) holds, with

e

T

jk

D 8T

jk

.Infact,(8.39)

implies that

e

R

`

D8,so

e

T

jk

D

e

R

jk

C 4g

jk

D 8



T

jk



1

2

g

jk

C

1

2

g

jk



D 8T

jk

:

Now, a computation parallel to that yielding (8.17) shows that in the present case,

(8.48) 

`

ˇ

S

D 0 H) 2G

0

k

ˇ

S

D

ı

g

k`

@

0



`

ˇ

S

C 2

e

T

0

k

:

Hence, if (8.47) holds, the hypothesis 

`

D 0 on S yields

(8.49) 

`

ˇ

S

D 0; @

0



`

ˇ

S

D 0:

Also, by (8.46), we have

e

T

jk

Ik

D 0,so(8.13)gives

(8.50) @

k

@

k



j

C B

j`

m

.g; rg/@

`



m

D 0;

and the initial-value problem (8.49)–(8.50) has only 

`

D 0 as a solution, so the

lemma is proved.

From here, one obtains the following parallel to Proposition 8.3:

Proposition 8.8. Suppose the initial data in (8.40)–(8.41) satisfy the consistency

conditions (8.44) and (8.47). Then there is a solution to

G

jk

D 8T

jk

satisfying these initial conditions, where T

jk

is the electromagnetic stress-energy

tensor, given by (8.38).

We next consider Einstein’s equations coupled with the equations of ﬂuid

motion. We use the form (6.59) of these equations, namely,

(8.51) r

w

.  2ˆ

0

r

2

w;w

/ Qw 

e

B.D

3

g; D

2

w; r/ D 0

and

(8.52) L

w

 D 0:

686 18. Einstein’s Equations

As in (8.37), we write Einstein’s equations as

(8.53) 

1

2

g

`m

@

`

@

m

g

jk

C H

jk

.g; rg/ D 8



T

jk



1

2

g

jk



;

where  D T

j

and this time

(8.54) T

jk

D . C p/u

j

u

k

C pg

jk

is the stress-energy tensor for a ﬂuid with 4-velocity u, density , and pressure p.

As in (6.37),

(8.55) w D e

q

u;dqD

dp

 C p

;

and Qw is the 1-form associated with w.Sincewewant(8.51)–(8.53) to be a system

of equations for .w;;g/, let us rewrite (8.54)as

(8.56) T

jk

D

 C p

hw; wi

w

j

w

k

C pg

jk

;D 



hw; wi



;pD p./:

The formula for  as a function of hw; wi follows implicitly from (8.55), since

e

2q

Dhw; wi.

We have made a slight notational change from (6.59)to(8.51), recording

the dependence of

e

B on D

3

g. Clearly, the coefﬁcients of the operator r

w

. 

2ˆ

0

r

2

w;w

/ also depend on D

3

g. Recall that

(8.57) ˆ

0

D



0

.p/  1

2e

2q

:

Also, as long as the equation of state satisﬁes 

0

.p/  1,(8.51) is strictly hyper-

bolic, and any hypersurface that is spacelike for the metric g

jk

is also spacelike

for (8.51).

Let us pose initial data on a compact hypersurface S, including the data (8.2),

with

(8.58)

ı

g

jk

2 H

`C2

.S/;

ı

k

jk

2 H

`C1

.S/:

We assume that S is spacelike for these data and that `>7=2. We also specify

(8.59) w

j

ˇ

S

2 H

`C1

.S/; @

0

w

j

ˇ

S

2 H

`

.S/; @

2

0

w

j

ˇ

S

2 H

`1

.S/;

and

(8.60) 

jk

ˇ

S

2 H

`

.S/:

9. Geometry of initial surfaces 687

Of course, there will be compatibility conditions that we will ultimately want to

place on such data, as discussed below, but these are not needed for the solvability

of (8.51)–(8.53). We will assume that w

ˇ

S

is timelike:

(8.61) hw; wi

ˇ

S

C

0

<0:

We identify a neighborhood of S with I S; I D ."; "/. Using the methods

of Chap. 16, in a fashion parallel to our discussion of (8.37)–(8.41), we obtain a

solution to (8.51)–(8.53) satisfying (8.58)–(8.60)and

(8.62)

g 2 C



I;H

`C2

.S/



\ C

1



I;H

`C1

.S/



;

w 2 C



I;H

`C1

.S/



\ C

2



I;H

`1

.S/



;

 2 C



I;H

`

.S/



:

We leave to the reader the demonstration that appropriate consistency condi-

tions on the initial data then imply that the Euler equations (6.4) are satisﬁed.

This in turn implies T

jk

Ik

D 0, and hence Lemma 8.7 is applicable. From there

one proceeds as before to show that when the initial data satisfy the consistency

conditions, one has a solution to (6.1)–(6.2).

As in the case of nonrelativistic (compressible) ﬂuids, one also considers the

phenomenon of shock waves in relativistic ﬂuids. For some work on this, see

[Lich3,Lich4, ST, ST2, Tau2].

Exercises

1. Write out the principal symbol L.x; / of the operator L in (8.3), and verify that

det L.x; / D 0;

for all  2 R

4

n 0.

2. Show that under appropriate consistency conditions on the initial data, solutions to

(8.51)–(8.53) also satisfy the Euler equations (6.4). (Hint: For one approach, see

[CBr3].)

3. Discuss the appropriate initial-value problem for the relativistic motion of a charged

ﬂuid, coupled both to the metric of spacetime and to an electromagnetic ﬁeld. The

resulting equations are the equations of relativistic magnetohydrodynamics.

Material on this can be found in [Lich3].

4. Make use of ﬁnite propagation speed to eliminate the hypothesis that the initial surface

S be compact, in the results of this section.

9. Geometry of initial surfaces

It is of interest to consider further when the initial data (8.2) satisfy the consistency

condition (8.20). When

ı

g

jk

is restricted to TS, we get a Riemannian metric on

SI h

jk

D

ı

g

jk

,for1  j  3. Let us assume for now that the coordinate

Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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